The complexity of classifying continuous t-norms up to isomorphism

It is shown that the isomorphism relation between continuous t-norms is Borel bireducible with the relation of order isomorphism between linear orders on the set of natural numbers, and therefore, it is a Borel complete equivalence relation.

💡 Research Summary

The paper investigates the classification problem for continuous t‑norms—binary operations on the unit interval that make it a commutative ordered monoid—and determines the descriptive‑set‑theoretic complexity of their isomorphism relation. After recalling that every continuous t‑norm can be uniquely expressed as an ordinal sum of three basic t‑norms (minimum, product, and Łukasiewicz), the authors encode each t‑norm by a countable family of disjoint open intervals together with a label indicating which basic t‑norm each interval realizes. This yields a map Υ from the space T of all continuous t‑norms to a set S of interval families equipped with a natural linear order ≺.

The central structural theorem (Theorem 2.6) shows that two continuous t‑norms are isomorphic precisely when there exists an order‑preserving bijection between their interval families that respects the three labels (product, Łukasiewicz, minimum). Consequently, the isomorphism problem is reduced to the problem of classifying countable labeled linear orders.

To analyze the complexity, the authors first endow the space C of all continuous binary functions on